231 Tutorial Sheet 512
11 November 2005
Useful facts:
• To evaluate the line integral for a parameterized curve:
Z
Z t2
dr
F · dt
F · dl =
dt
t1
c
(1)
where t1 and t2 are the parameter values corresponding to the begining and end of
the curve.
• To evaluate the surface integral for a parameterized surface:
Z Z
Z Z
dr dr
F · dA =
F·
dudv
×
du dv
S
S
(2)
Questions
1. (a) Compute the flux of the vector field F = xi+2yj+zk out of the cylinder defined
by x2 + y 2 = 1 and 0 ≤ z ≤ 1.
(b) What is the flux if the vector field is replaced with F = r = xi + yj + zk?
(c) Find the flux of F = z 2 k upwards through the part of the sphere x2 +y 2 +z 2 = a2
in the first octant of R3 (in the first octant all three coordinates x, y and z are
positive).
2. Find the flux of F = 2xi + yj + zk upwards through the surface with parametrization
r(u, v) = u2 vi + uv 2 j + v 3 k where u and v range from 0 to 1.
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Conor Houghton, houghton@maths.tcd.ie, see also http://www.maths.tcd.ie/~houghton/231
Including material from Chris Ford, to whom many thanks.
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